Hard core phonon frequency at transition temperature

A simple theoretical derivation is given to show that one may expect a hard core for the phonon frequency at the transition temperature. This approach implies that the frequency of the phonon will follow $\text{ω ∝ {[1 + γ(T ? T}{}_{\mathrm{<mtext>C</mtext>}}\text{)] V}{}_0\text{}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ instead of that predicted by soft mode theory. Our expression should be valid whenever a Brownian sublattice exists in the crystal.